The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

Abstract

The viscoelastic analogue to the Newtonian Orr amplification mechanism is examined using linear theory. A weak, two-dimensional Gaussian vortex is superposed onto a uniform viscoelastic shear flow. Whilst in the Newtonian solution the spanwise vorticity perturbations are simply advected, the viscoelastic behaviour is markedly different. When the polymer relaxation rate is much slower than the rate of deformation by the shear, the vortex splits into a new pair of co-rotating but counter-propagating vortices. Furthermore, the disturbance exhibits a significant amplification in its spanwise vorticity as it is tilted forward by the shear. Asymptotic solutions for an Oldroyd-B fluid in the limits of high and low elasticity isolate and explain these two effects. The splitting of the vortex is a manifestation of vorticity wave propagation along the tensioned mean-flow streamlines, while the spanwise vorticity growth is driven by the amplification of a polymer torque perturbation. The analysis explicitly demonstrates that the polymer torque amplifies as the disturbance becomes aligned with the shear. This behaviour is opposite to the Orr mechanism for energy amplification in Newtonian flows, and is therefore labelled a 'reverse-Orr' mechanism. Numerical evaluations of vortex evolutions using the more realistic FENE-P model, which takes into account the finite extensibility of the polymer chains, show the same qualitative behaviour. However, a new form of stress perturbation is established in regions where the polymer is significantly stretched, and results in an earlier onset of decay.

Critical-Layer Structures and Mechanisms in Elastoinertial Turbulence

“…In both analyses, the concept of critical layers, i.e., wall-normal positions where the fluid velocity equals the wavespeed of an eigenmode or resolvent mode, is important. While some recent studies suggest the importance of critical-layer mechanisms in viscoelastic shear flows [12,[15][16][17], they do not make as direct a connection to EIT as we illustrate here. Figure 3a shows the result of linear stability analysis (the eigenvalues c) for Wi = 20, k x L x /2π = 2, k z = 0, the wavenumber corresponding to the dominant structures observed in the nonlinear simulations.…”

Simulations of natural transition in viscoelastic channel flow

“…The sheet itself is due to the primary Tollmien-Schlichting wave, which is a spanwise vorticity perturbation superimposed onto the mean shear. This connection was explained by Page & Zaki (2015) in the more canonical setting of a spanwise Gaussian vortex in homogeneous shear. Two key effects lead to this phenomenology: (i) the action of the spanwise vorticity perturbation onto the large mean C xx and (ii) the action of the mean shear on the perturbation in the polymer conformation (for details see Page & Zaki 2015).…”

“…i ⌘ ✏ ijk @f 0 k /@x j where f 0 k = @⌧ 0 km /@x m . It is directly proportional to the toque exerted on a fluid element by the polymer forces (Page & Zaki 2015). In the streamwise direction, the evolution equation for !…”

Inertioelastic Flow Instability at a Stagnation Point

“…14(c)]. It is already well known that the polymer torque resulting in such a situation acts counter to the vorticity [92][93][94][95][96], which explains why the vorticity growth is suppressed in the weakly elastic fluids compared with the Newtonian case as Re is increased above Re c . Such a mechanism is thought to be responsible for suppression of streamwise and hairpin vortices in polymer drag-reduced flows [92,93].…”